371
ACTA MATHEMATICA VIETNAMICA Volume 34, Number 3, 2009, pp. 371–374
A NOTE ON THE FINITENESS PROPERTY RELATED TO DERIVED FUNCTORS AMIR MAFI AND HERO SAREMI
Abstract. Let R be a commutative Noetherian ring, a an ideal of R, and M , N two finitely generated R-modules. Let t be a non-negative integer. It is shown that for any finitely generated R-module L with Supp(L) ⊆ Supp(M ), the following statements hold: (i) Supp(ExttR (L, N )) ⊆ ∪ti=0 Supp(ExtiR (M, N )); i (ii) Ass(ExttR (L, N )) ⊆ Ass(ExttR (M, N )) ∪ (∪t−1 i=0 Supp(ExtR (M, N ))). As an immediate consequence, we deduce that if Supp(Hai (N )) or Supp(Hai (M, N )) is finite for all i < t, then the set ∪n∈N Ass(ExttR (M/an M , N )) is finite. In particular, if grade(a, N ) ≥ t then the set ∪n∈N Ass(ExttR (M/an M , N )) is finite.
1. Introduction Throughout this note, we assume that R is a commutative Noetherian ring with non-zero identity, a an ideal of R, and that M and N two finitely generated R-modules. We use N to denote the set of positive integers. Brodmann [1] proved that the two sequences of associated primes (Ass(N/an N ))n∈N and (Ass(an N/an+1 N ))n∈N eventually become constant for large n. Melkersson and Schenzel [10] showed that, for any given integer i ≥ 0, the sequences n (Ass(TorR i (R/a , N )))n∈N
and
(Att(ExtiR (R/an , A)))n∈N
become, for n large, independent of n where A is an Artinian R-module. They also asked whether the set Ass(ExtiR (R/an , N )) becomes stable for sufficiently large n. Khashyarmanesh and Salarian [7], gave an affirmative answer to the above question in the case i = 1. Katzman [5] gave an example of a Noetherian local ring (R, m) with two elements x, y ∈ m such that the associated prime 2 ideals of local cohomology module H(x,y) (R) is an infinite set. Therefore the set n 2 ∪n∈N Ass(ExtR (R/(x, y) , R)) is infinite and so ∪n∈N Ass(ExtiR (R/an , N )) is not a finite set in general. Received January 18, 2008. 2000 Mathematics Subject Classification. 13D45, 13E99. Key words and phrases. Local cohomology modules, generalized local cohomology modules, support of local cohomology modules, associated prime ideals.
372
A. MAFI AND H. SAREMI
For an integer i ≥ 0, the i-th generalized local cohomology module Hai (M, N ) of two R-modules M and N with respect to an ideal a is defined by Hai (M, N ) = lim ExtiR (M/an M , N ). −→ n
Hai (R, N )
It is clear that is just the ordinary local cohomology module Hai (N ) of N with respect to a. We refer the reader to [2] and [4] for the basic properties of local cohomology and generalized local cohomology. The aim of this note is to prove the following theorems. Theorem 1.1. Let t be a non-negative integer. Then for any finitely generated module L with Supp(L) ⊆ Supp(M ), the following statements hold: (i) Supp(ExttR (L, N )) ⊆ ∪ti=0 Supp(ExtiR (M, N )); i (ii) Ass(ExttR (L, N )) ⊆ Ass(ExttR (M, N )) ∪ (∪t−1 i=0 Supp(ExtR (M, N ))). Theorem 1.2. Let t be a non-negative integer such that Supp(Hai (N )) or Supp(Hai (M, N )) is finite for all i < t. Then the set ∪n∈N Ass(ExttR (M/an M , N )) is finite. In particular, if grade(a, N ) ≥ t then ∪n∈N Ass(ExttR (M/an M , N )) is finite. 2. The results Theorem 2.1. Let t be a non-negative integer. Then for any finitely generated module L with Supp(L) ⊆ Supp(M ), the following statements hold: (i) Supp(ExttR (L, N )) ⊆ ∪ti=0 Supp(ExtiR (M, N )); i (ii) AssR (ExttR (L, N )) ⊆ Ass(ExttR (M, N )) ∪ (∪t−1 i=0 Supp(ExtR (M, N ))). Proof. We will only prove the first part, and the proof of the second part is similar. We use induction on t. Let t = 0. Since Supp(L) ⊆ Supp(M ), we get by Gruson’s Theorem [11, Theorem 4.1] that there exists a finite filtration 0 = L0 ⊂ L1 ⊂ . . . ⊂ Lk = L such that the factors Li /Li−1 are homomorphic images of a direct sum of finitely copies of M . By using short exact sequences, we may reduce the situation to the case k = 1. Then there is an exact sequence 0 −→ K −→ M m −→ L −→ 0, for some m ∈ N and some finitely generated R-module K. This induces an exact sequence 0 −→ HomR (L, N ) −→ HomR (M m , N ) and so the result for t = 0 is complete. Now suppose, inductively, that t > 0 and we have established that Supp(ExtjR (L, N )) ⊆ ∪ji=0 Supp(ExtiR (M, N )) for all j < t and all finitely generated modules L with Supp(L) ⊆ Supp(M ). Again, by using Gruson’s Theorem, we have an exact sequence 0 −→ K −→ M m −→ L −→ 0 that induces a long exact sequence t−1 . . . −→ ExtR (K, N ) −→ ExttR (L, N ) −→ ExttR (M m , N ) −→ . . . .
THE FINITENESS PROPERTY RELATED TO DERIVED FUNCTORS
373
Therefore Supp(ExttR (L, N )) ⊆ Supp(ExttR (M, N )) ∪ Supp(Extt−1 R (K, N )) and so by inductive hypothesis the result follows. The following corollary immediately follows by Theorem 2.1. Corollary 2.2. Let t be a non-negative integer. Then for any finitely generated module L with Supp(L) = Supp(M ), the following statements hold: (i) ∪ti=0 Supp(ExtiR (L, N )) = ∪ti=0 Supp(ExtiR (M, N )); (ii) i Ass(ExttR (L, N )) ∪ (∪t−1 i=0 Supp(ExtR (M, N ))) i = Ass(ExttR (M, N )) ∪ (∪t−1 i=0 Supp(ExtR (M, N ))).
Corollary 2.3. Let t be a non-negative integer. Then the following equalities holds. ∪ti=0 (∪n∈N Supp(ExtiR (M/an M , N ))) = ∪ti=0 Supp(ExtiR (M/aM , N )) = ∪ti=0 Supp(Hai (M, N )). Proof. The first equality follows from Corollary 2.2 and the second equality follows from [3, Lemma 2.8]. The following theorem extends [6, Theorem 2.4], [8, Theorem B] and [9, Theorem 2.12]. Theorem 2.4. Let t be a non-negative integer. Then the following statements hold: (i) If Supp(Hai (N )) is finite for all i < t, then the set ∪n∈N Ass(ExttR (M/an M , N )) is finite. In particular, Ass(Hat (N )) and Ass(Hat (M, N )) are finite. (ii) If Supp(Hai (M, N )) is finite for all i < t, then the set ∪n∈N Ass(ExttR (M/an M , N )) is finite. In particular, Ass(Hat (M, N )) is finite. Proof. Apply Theorem 2.1 and Corollary 2.3.
The following corollary immediately follows by Theorem 2.4. Corollary 2.5. Let grade(a, N ) ≥ t. Then the set ∪n∈N Ass(ExttR (M/an M , N )) is finite. In particular, Ass(Hat (N )) and Ass(Hat (M, N )) are finite. The following result extends [7, Corollary 2.3]. Corollary 2.6. The set ∪n∈N Ass(Ext1R (M/an M , N )) is finite. In particular, Ass(Ha1 (N )) and Ass(Ha1 (M, N )) are finite. Proof. The exact sequence 0 −→ Γa (N ) −→ N −→ N/Γa (N ) −→ 0 provides an exact sequence 0 −→ Ext1R (M/an M , Γa (N )) −→ Ext1R (M/an M , N ) −→ Ext1R (M/an M , N/Γa (N )). Thus ∪n∈N Ass(Ext1R (M/an M , N )) ⊆ (∪n∈N Ass(Ext1R (M/an M , Γa (N )))) ∪ (∪n∈N Ass(Ext1R (M/an M , N/Γa (N )))). On the other hand, by Corollary 2.5
374
A. MAFI AND H. SAREMI
∪n∈N Ass(Ext1R (M/an M , N/Γa (N ))) is finite and so it is enough to prove that ∪n∈N Ass(Ext1R (M/an M , Γa (N ))) is finite. The exact sequence 0 −→ an M −→ M −→ M/an M −→ 0 induces, for large n, the following exact sequence 0 −→ HomR (an M, Γa (N )) −→ Ext1R (M/an M , Γa (N )) −→ Ext1R (M, Γa (N )). This proves that ∪n∈N Ass(Ext1R (M/an M , Γa (N ))) is finite, as required.
References [1] M. P. Brodmann, Asymptotic stability of Ass(M/I n M ), Proc. Amer. Math. Soc. 74 (1) (1979), 16–18. [2] M. P. Brodmann and R. Y. Sharp, Local Cohomology-an Algebraic Introduction with Geometric Applications, Cambridge University Press, 1998. [3] N. T. Cuong and N. V. Hoang, On the vanishing and the finiteness of supports of generalized local cohomology modules, Manuscripta Math. 126 (1) (2008), 59–72. [4] J. Herzog, Komplexe, aufl¨ osungen und dualitat in der localen algebra, Habilitationss chrift, Universit¨ at Regensburg, 1970. [5] M. Katzman, An example of an infinite set of associated primes of a local cohomology module, J. Algebra 252 (2002), 161–166. i [6] K. Khashyarmanesh, On the finiteness properties of ∪i AssR Extn R (R/a , M ), Comm. Algebra 34 (2) (2006), 779–784. n [7] K. Khashyarmanesh and Sh. Salarian, Asymptotic stability of AttR TorR 1 ((R/a ), A), Proc. Edinb. Math. Soc. 44 (2001), 479–483. [8] K. Khashyarmanesh and Sh. Salarian, On the associated primes of local cohomology modules, Comm. Algebra 27 (12) (1999), 6191–6198. [9] A. Mafi, On the finiteness results of the generalized local cohomology modules, Algebra Colloquium 16 (2) (2009), 325–332. [10] L. Melkersson and P. Schenzel, Asymptotic prime ideals related to derived functors, Proc. Amer. Math. Soc. 117 (4) (1993), 935–938. [11] W. Vasconcelos, Divisor Theory in Module Categories, North-Holland, Amsterdam, 1974. Department of Mathematics, University of Kurdistan, P.O. Box: 416, Sanandaj, Iran E-mail address: a
[email protected] Azad University of Sanandaj, Pasdaran St., P.O. Box 618, Kurdistan, Iran E-mail address:
[email protected]
